Let \(S\)  and \(\Gamma \)  be as in the statement, and let \({C}\)  be the Cantor set that is the unique equivalence class in \(\mathcal {M}(E)\) . Let \(c\)  be a curve in \(\Gamma \) . Since \(S)=1\) , we have \(\text{g}(S)=0\)  or \(\infty \) . If \(c\)  is a nonseparating curve, then \(\text{g}(S)=\infty \) , Proposition REF (3) applies, and \(\Gamma \)  is not good.
If \(c\)  is separating and cuts off a finite-type surface, then \(\Gamma \)  is not good since either Proposition REF (1) or (2) applies, as follows. Since we already have that \(\text{g}(S)=0\)  or \(\infty \) , we next show that \(S\)  has either 0 or infinitely many isolated punctures. The surface \(S\)  cannot have one or more (but finitely-many) isolated punctures, since these would be maximal, contrary to the hypothesis on \(\mathcal {M}(E)\) . Finally, if \(S\)  has infinite genus and infinitely many isolated punctures, then each end in \(\mathcal {M}(E)\)  is accumulated both by genus and by isolated punctures, and so we conclude that the proposition applies.
Otherwise, \(c\)  is separating but does not cut off a finite-type subsurface. In Proposition REF  there was only one case to consider, but here there are two: either only one component of \(S \setminus c\)  has end space with nontrivial intersection with \({C}\) , or instead both do. The former case is analogous to the only case in Proposition REF , and the same argument holds here as in that proposition, substituting \({C}\)  for \(e\) . In this case, then, \(\Gamma \)  is not good.
The remaining case where both components of \(S \setminus c\)  have end spaces with nontrivial intersection with \({C}\)  follows similarly to the treatment of the Cantor tree surface by Durham–Fanoni–Vlamis [1]}. Since \(S\)  has a unique maximal equivalence class of ends, any clopen neighborhood in \(E(S)\)  of any maximal end is homeomorphic to \(E(S)\) . In particular, the end spaces of the two components of \(S \setminus c\)  are each homeomorphic to \(E(S)\) . Set
\( \mathcal {V} = \lbrace (a, b)~|~i(a, b) = 0, \ a \text{ and } b \text{ are distinct.}\rbrace \) 
By the classification of surfaces and the self-similarity of \(E(S)\) , there is a unique pair of distinct disjoint separating curves \((a,b) \in \mathcal {V}\)  with \(a, b \in \operatorname{Map}(S) \cdot c\) , up to the action of \(\operatorname{Map}(S)\) . Therefore condition (1) of Proposition REF  is satisfied.
Next, consider \(a, b \in \operatorname{Map}(S) \cdot c\)  such that \((a, b) \notin \mathcal {V}\) . Then either \(a\)  and \(b\)  intersect or else they are not distinct. In the former case, \(a\)  and \(b\)  and together they fill a finite-type subsurface \(F\) . At least one component of \(S \setminus \partial F\)  has end space homeomorphic to \(E(S)\)  by Lemma REF . Then there exists a separating curve \(d\)  in this complementary component that is in \(\operatorname{Map}(S) \cdot c\)  and that is distinct from and disjoint from each of \(a\)  and \(b\) . We then have \((a,d), (b,d) \in \mathcal {V}\) . A similar argument applies if \(a\)  and \(b\)  are not distinct. Condition (2) of Proposition REF  is therefore satisfied, and \(\Gamma \)  is not good.
Since this exhausts the possibilities for \(c\) , we conclude that \(S\)  has no good curve graph.
